An efficient EM algorithm for the mixture of negative binomial models

Overdispersion is a widespread phenomenon in most count data sets. The negative binomial distribution is commonly adopted to fit over-dispersed count data. On the other hand, the mixture model always plays an important role in unsupervised classification. However, when estimating the parameters in the mixture of negative binomial models, the typical generalized Expectation Maximization (EM) algorithm which involves additional iterative procedures in M-step increases computational time. Hence, there remains a need for an efficient algorithm that can speed up the procedure of parameter estimation. For this purpose, here we develop a novel EM algorithm that successfully avoids the typical numerical solution in M-step for the mixture of negative binomial models. We extend further this EM algorithm to the zero-inflated negative binomial model. In the simulation studies, we focus on the runtimes and the classification performance of our proposed algorithm implemented in the mixture of negative binomial model. We found that our proposed EM algorithm can reduce the runtime of maximum likelihood estimation effectively, while achieving the similar classification performance in comparison with the typical EM algorithm. The mixture of negative binomial model and the proposed EM algorithm finally illustrates their good performance of fitting the real earthquake count data.